Define the Frobenius norm of a matrix as $\left\Vert A \right\Vert_{\mathrm{F}}=\sqrt{\sum_{i,j} A_{ij}^2}$ and the operator norm as $\left\Vert A \right\Vert_{\mathrm{op}}=\sup_{x \not = 0} \frac{\left\Vert Ax\right\Vert_2}{\left\Vert x \right\Vert_2}$ where the the norm in the numerator and denominator are the standard Euclidean norm.

If $\mathbf{N}$ and $\mathbf{M}$ are normal matrices on a separable complex Hilbert space $H$, and $f$ is a Lipschitz function defined on the spectrum of both matrices $\Omega = \sigma(\mathbf{N}) \cup \sigma(\mathbf{M}) $ with Lipschitz constant $k$ then $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_\mathrm{F} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_\mathrm{F}$. This is a result from Kittaneh (1985). Note that in this paper they use the notation $\left\Vert \cdot \right\Vert_2$ to be the Hilbert–Schmidt operator which I believe is the Frobenius norm in the finite dimensional case.

I found this recent survey paper on operator Lipschitz functions which states similar results. From what I can tell the norm isn't specified but I think the results are also for the Frobenius norm.

I would like to know

- Is the term "operator Lipschitz function" reserved for functions which have the property under the Frobenius norm, or is it a more general concept defined for any norms?
- Does the result from Kittaneh (1985) hold for operator norms and if so what is the reference? (The proof in this paper seems specific to the Frobenius norm). The review paper says for equation 3.1.2 "It follows easily from the spectral theory for pairs commuting normal operators", I'm not quite sure what this is and if it holds for norms generally (I haven't had much much formal training in functional analysis or measure theory)

Specifically for my research I have real symmetric matrices $\mathbf{N}$ and $\mathbf{M}$ with eigenvalues lying in the interval $[-1, 1]$. If I have a Lipschitz continuous function $f:[-1,1] \rightarrow \mathbb{R}$ (we can also add differentiability or infinitely differentiability as an assumption if it helps) does it hold that $\left\Vert f(\mathbf{N}) - f(\mathbf{M})\right\Vert_{\mathrm{op}} \leq k\left\Vert \mathbf{N} - \mathbf{M}\right\Vert_{\mathrm{op}}$ where $k$ is the Lipschitz constant of $f$?

I hope this question isn't too basic for MO, I asked on the mathematics SE but didn't get a response.